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Abstract

We apply Kirchhoff’s heat equation to model the influence of a CW terahertz beam on a sample of water, which is assumed to be static. We develop a generalized model, which easily can be applied to other liquids and solids by changing the material constants. If the terahertz light source is focused down to a spot with a diameter of 0.5 mm, we find that the steady-state temperature increase per milliwatt of transmitted power is 1.8°C/mW. A quantum cascade laser can produce a CW beam in the order of several milliwatts and this motivates the need to estimate the effect of beam power on the sample temperature. For THz time domain systems, we indicate how to use our model as a worst-case approximation based on the beam average power. It turns out that THz pulses created from photoconductive antennas give a negligible increase in temperature. As biotissue contains a high water content, this leads to a discussion of worst-case predictions for THz heating of the human body in order to motivate future detailed study. An open source Matlab implementation of our model is freely available for use at www.eleceng.adelaide.edu.au/thz.

Figures (7)

Illustration of the cross-section of the disc of water, which is used in the calculations. Here, b and d are the radius and the thickness of the disc, respectively, a is the radius of the THz beam and [a], [b], [c] and [d] denote the chosen boundary conditions for the transformed temperature, U (r, z). The calculations are performed in cylindrical coordinates, which is why only half of the cross-section is needed by argument from symmetry.

The temperature change per milliwatt, Ũ(r,z), (a) at z = 0 as a function of the radial distance, r, and (b) at r = 0 as a function of the thickness, z. It is seen that the maximal heating occurs just in the center of the beam and that the result has an exponential tail, but this may vary depending on the choice of dissipated power.

(a) The temperature change per milliwatt at (r,z) = (0,0) as a function of the number of terms used in the sum. After 400 terms it appears that the value of Ũ(r,z) is saturated - but to increase the accuracy, 1000 terms will be used in all further calculations. (b) The temperature change at (r,z) = (0,0) as a function of the sample radius, b, where the dashed red line indicates the chosen initial value. The result is shown for the three cases where boundary conditions b.1 and b.2 are used and when Cn is neglected. The last case makes the result independent of which boundary condition is used. It is seen that the dependence of Ũ(r,z) on the radius of the disc is negligible and that the choice of boundary conditions at the side of the disc is unimportant when b > 25 mm. For smaller b the three results are very different, which makes the result less reliable and therefore we choose b = 50mm.

The dashed red line on both graphs indicates the chosen initial values. (a) The temperature change per milliwatt at (r,z) = (0,0) as a function of the sample thickness, d. If the chosen thickness of the water disc is too small, the profile along the z-axis loses its exponential behavior, which is unphysical. Thus we have chosen a thickness of 15 mm for the sake of example. (b) The temperature change vs. the beam radius, a, at (r,z) = (0,0). It is hard to focus a THz beam with a center frequency of 1 THz down to a spot radius smaller than 0.25 mm [20] due to the diffraction limit, and most often the radius will be around 0.5 mm. Note that this quantity is investigated even further in Fig. 5.

(a)-(f) Contour plots, with the normalized radius, r/a, on the x-axis and the thickness, z on the y-axis showing the heating caused by the THz beam for six different beam radii a = 0.05mm, a = 0.25mm, a = 0.75mm, a = 2.50mm, a = 5.00mm and a = 10.00mm. The colors indicate the level of heating and the color-bar above each plot shows the conversion (in Kelvin per milliwatt), the vertical dashed lines show where the beam hits the water. (g)-(h) The normalized temperature change, Ũ(r,z)/Ũmax, as a function of the normalized radius, r/a, and the thickness z, respectively. The six curves represent the six different beam radii as shown on the contour plots.

(a) The temperature increase at (r,z) = (0,0) of the initial temperature, T0. (b) The temperature increase at (r,z) = (0,0) as a function of the beam frequency, ν. The red dashed line on each plot indicates the chosen initial value.

Contour plot of the sample near the center, the colors indicate the temperature change per milliwatt, Ũ(r,z), and the vertical lines show where the terahertz beam hits the disc. (b) The power needed to create a temperature change of 1 K as a function of the beam radius. The red vertical dashed line indicates the chosen initial value for the beam radius and the horizontal line shows the power corresponding to this value (0.56 mW). Note that P and U are directly proportional (P = U/Ũ(r,z)), i.e with a beam radius of 0.25 mm the amount of transmitted power needed to heat the water for instance 2°C is 2 × 0.56mW = 1.2mW. The initial conditions given in Table 2 is used in both figures.